Bob Wentworth Ph.D. (Applied Physics)
Recently, Stephen Wilde invited me to “have a go at deconstructing” the work he and Philip Mulholland have been doing to understand how climate functions. I was curious. So, I began looking at what Wilde and Mulholland (W&M) have written.
Today, I’d like to examine a building block concept which impacts their work, energy recycling.
It’s a topic that leads to seemingly endless confusion among people who doubt that long-wave-absorbing gases can warm planets. So, this topic is likely to be of interest beyond its relevance to W&M’s work.
This inquiry was stimulated by reading W&M’s 2020 paper, An Analysis of the Earth’s Energy Budget. W&M’s analysis is informed in part by the diagram below (my Figure 1, W&M Figure 4, originally from Oklahoma Climatological Survey).
This figure illustrates how a layer of the Earth’s atmosphere interacts with radiant energy.
Solar short-wave radiation, with a mean radiant flux, Fₛ(1-A)/4, is absorbed by the Earth’s surface. The Earth’s surface, at an effective radiative temperature, T₀, emits long-wave thermal radiation with a flux, σT₀⁴, in accordance with the Stephan-Boltzmann law. A fraction (1-f) of this surface-emitted long-wave radiation passes through the atmosphere and reaches space, while a fraction f is absorbed by the atmosphere.
A particular layer of the atmosphere is assumed to be at a temperature, T₁. This temperature is the temperature that equalizes the flows of energy entering and leaving that layer. According to the diagram, the layer will emit long-wave radiant energy equally in all directions, with a flux fσT₁⁴ being sent upward and an equal flux being sent downward.
It should be noted that this diagram is intended for general education, and oversimplifies some details that a serious climate modeler would take into account. In particular, I see the following simplifications:
- The diagram depicts the total absorbed mean solar irradiance, Fₛ(1-A)/4, being absorbed by the Earth’s surface. However, something like 27% of that is actually absorbed into the atmosphere (via clouds, water vapor, dust, and ozone).
- The long-wave flux emitted by the Earth’s surface actually has the form 𝜀₀σT₀⁴, where 𝜀₀ is the mean emissivity of the surface, which has been measured to be 0.94.
- How much long-wave radiation is emitted by a layer of the atmosphere depends on the thickness of that layer. Saying the radiant flux is fσT₁⁴ reflects a few implicit assumptions, namely that (a) the layer has sufficient optical depth that it absorbs most of the incident radiative at the wavelengths of interest and (b) the temperature doesn’t vary much across the layer. Serious modeling would involve formulas for the radiative properties of a thin layer of atmosphere, as well as accounting for convection, etc.
- The radiant flux emitted by an atmospheric layer is given as fσT₁⁴, but would more accurately be given by 𝜀σT₁⁴ where 𝜀 is the emissivity of the gas. It’s likely to be approximately true that 𝜀 ≈ f, but this may not be precisely the case. Additionally, the overall emissivity of a gas depends somewhat on temperature, so the radiated flux may not scale precisely as T₁⁴.
Yet, the purpose of the diagram is public education, not rigorous modeling. For that purpose, the diagram has its uses.
How do W&M apply this diagram? In part, they correctly note (p. 56) that some “of all captured flux is returned to the surface as back radiation and recycled.” (More precisely, they assume that “half” of the flux captured by the atmosphere is returned to the surface; that’s not quite right, but we’ll return to this point later.) They also note (p. 57) that “Because the intercepted energy flux is being recycled this feed-back loop is… endless … It has the mathematical form of a geometric series, and is a sum of the descending fractions…”
Let’s look at a diagram that illustrates the energy recycling process that W&M are talking about.
In this diagram, sunlight with power S is absorbed by the surface of the Earth. (For simplicity in sorting out the concepts, we’ll ignore the solar irradiation that is directly absorbed by the atmosphere.)
Because the surface of the Earth is assumed to be neither gaining or losing net energy (when averaged over a day or a year), the amount of power absorbed by the surface must lead to an equal amount of energy leaving the surface. The power leaves the surface via a combination of thermal radiation and convective transport of latent heat (water vapor) and sensible heat (hot air).
Suppose we assume that, for every unit of energy flux that leaves the surface, a fraction (1-β) is radiated into space, and the remaining fraction, β, is returned to the surface via long-wave back-radiation. (In steady-state, on average, the energy flux leaving the atmosphere must equal the energy flux entering the atmosphere. Hence, any energy flux that doesn’t reach space must be returned to the surface, for energy flux balance to hold.)
For each energy flux that reaches the surface, an equal energy flux leaves the surface and enters the atmosphere. A fraction (1-β) reaches space, and a fraction β is returned to the surface. The energy flux returned to the surface must lead to an equal flux leaving. This results in another cycle of some energy reaching space, and some being returned to the surface. In principle, this recycling continues forever, with ever smaller fluxes. Because each round of the cycle reduces the flux by a fixed proportion, the fluxes form a geometric series, making it easy to sum the infinite series. Computing these sums, one finds that the total power radiated into space is S, the same as the energy flux absorbed by the Earth. That’s as one would expect.
One finds that the total back-radiated energy flux, B, is given by B = β⋅S/(1-β).
Climate models don’t usually include a figure like Figure 2 above, in which each iteration of the energy recycling process is shown. Diagrams like Figure 2 are useful for instructional purposes, but aren’t as practical as other ways of depicting things.
Instead, climate models often offer a diagram of total energy fluxes, like the one below. This diagram shows the net result, after all the recycled energy flows have been added together.
This diagram shows solar flux, S, being absorbed by the surface. There is also an energy flux S/(1-β) leaving the surface (via thermal radiation and other heat transfer), and a back-radiation energy flux B = β⋅S/(1-β) from the atmosphere to the surface. The radiant energy flux leaving the top of the atmosphere is S, equaling the amount of solar irradiance that was absorbed by the Earth. The thickness of the lines qualitatively suggests the differing magnitudes of these energy fluxes.
For Earth, the data in Kiehl and Trenberth (1997), which is used as a reference by W&M, indicate a ratio of back-radiation to absorbed insolation, B/S = 1.38. This corresponds to a recycling fraction β = 0.58. (These calculations pretend all absorbed solar irradiance is absorbed by the surface.)
Sometimes people are incredulous at the idea that the back-ration flux, B, is greater than the absorbed insolation, S. Yet, this is what is measured to be true.
The energy recirculation diagram, Figure 2, should make clear how this can and does happen, without requiring that anything “fishy” be going on.
It might be reassuring to look at heat flow, instead of the usual energy balance diagram (like Figure 3 above) which mixes heat flows with radiant energy flows. Recall that heat flow is the net energy flow, so that a heat flow (unlike an energy flow) is only in one direction. Translating Figure 3 into an equivalent heat flow diagram yields the diagram below.
If one takes the combined energy flux away from the surface, S/(1-β), and subtracts the back-radiation flux, β⋅S/(1-β), one finds that the heat flux from the surface to the atmosphere is S, exactly the same as the heat flux absorbed from the Sun, and the heat flux radiated into space.
There is nothing contrary to energy conservation happening here. It all adds up.
To some people, it seems counter-intuitive to some that energy recirculation can result in recirculating energy fluxes higher than the initiating absorbed energy flux. But, this result, while perhaps surprising, is not wrong. The math is quite straightforward, as I think I’ve shown.
* * *
Of course, having back-radiation be greater than the absorbed insolation requires that the recycling fraction, β, be larger than ½.
W&M assume that the largest β can get is ½, in which case the back-radiation flux is B = S. They write (p. 55):
“The standard assumption is that for all energy fluxes intercepted by the atmosphere, half of the flux is directed upwards and lost to space, and half of all captured flux is returned to the surface as back radiation and recycled.”
It appears that W&M reach their conclusion that this is the “standard assumption” by examining Figure 1 (their Figure 4), and noting that an atmospheric layer radiates an equal amount upward and downward.
The conclusion that equal amounts are radiated upward and downward is correct—but only for a single layer of the atmosphere.
The atmosphere has more than one layer. To consider the behavior of the atmosphere as a whole, one needs to consider the aggregate effect of many layers interacting with one another.
To illustrate this, let’s look at a “toy model” of the atmosphere, consisting of N layers, each of which behaves like the atmospheric layer in Figure 1.
Sunlight with an average flux, S, is absorbed by the planetary surface.
The surface emits a total flux of long-wave radiation, σT₀⁴. For simplicity, we assume a fraction (1-f) of this thermal radiation has wavelengths that pass through the atmosphere unhindered, while a fraction f is at wavelengths which are totally absorbed by each layer of the atmosphere.
Each layer of the atmosphere has a distinct temperature, and radiates equally in both directions, with a radiant flux fσT⁴.
For simplicity, I assume that only radiative heat transfer is relevant. I assume that the radiant long-wave flux from space is negligible.
This model is not a realistic representation of Earth’s atmosphere. But, solving this problem is likely to be informative, nonetheless.
Using energy balance, we can solve for all the temperatures. This is done easily using a corresponding heat flow diagram.
Here’s how the calculation works. Feel free to skip these details. I denote the heat flux from the surface to the atmosphere, Q. Because the heat flowing to and away from the surface must balance, we know Q=S-(1-f)σT₀⁴. Energy balance also tells us that the heat flowing to and from each atmospheric layer matches, so that Q flows between each layer, and out of the final layer. Comparing the amount flowing out of the final layer in Figure 5 and 6 allows us to solve for the temperature of the last layer, Tₙ, in terms of Q. That allows one to solve for the temperature of each layer in turn, finally yielding a formula for T₀ in terms of Q. Combining this with the previous formula for Q allows us to eliminate Q and solve for T₀ in terms of S and N.
The layers of the atmosphere have T⁴ values which are spaced linearly between the value for the surface, T₀⁴, and zero (the assumed value for space). In this model, the atmosphere gets monotonically colder as one moves to higher layers.
Other key results include:
Q = S⋅f/[1+N(1-f)]
T₀⁴ = (S/σ)⋅(N+1)/[1+N(1-f)]
B/S = N⋅f/[1+N(1-f)]
Relating this model to the prior energy-recycling model, one finds that the energy recycling fraction associated with N atmospheric layers is:
β = N⋅f/(N+1)
For an atmosphere that is opaque to long-wave radiation (i.e., f=1), then a single layer (N=1) yields β=½ and B = S, as assumed by W&M. However, in general, for an atmosphere opaque to long-wave radiation, the energy recycling fraction is β = N/(N+1) and back-radiation flux is B = N⋅S.
As long as an atmosphere has more than one layer, it is entirely possible for the recycling fraction to be greater than ½, and for the back-radiation flux to be arbitrarily large, compared to the absorbed insolation.
How can we make intuitive sense of this result?
An atmosphere as a whole is not at a single temperature.
In our “toy model”, the top layer is much colder than the bottom layer. The radiant flux downward to the surface (the “back radiation”) is determined by the temperature of the bottom layer. The radiant flux upward to space is determined by the temperature of the top layer. Because of the temperature difference between the top and bottom layers, it is entirely natural that the atmosphere as a whole directs more radiation downward to the surface than it does upwards to space.
* * *
Thus, what W&M interpreted as “the standard assumption” that “for all energy fluxes intercepted by the atmosphere, half of the flux is directed upwards and lost to space, and half of all captured flux is returned to the surface as back radiation and recycled” is false.
It’s not “the standard assumption” with regard to the atmosphere as a whole. It’s a false assumption for the atmosphere as a whole, as demonstrated by our model of a multi-layer atmosphere.
The hypothesis that the atmosphere as a whole behaves this way is also contradicted by measurements. Those measurements show significantly more flux being directed downwards to the surface than is directed upwards and lost to space (by a factor of around 1.38).
* * *
Let’s use the results of our modeling to think through a few issues unrelated to W&M’s work.
Does the model violate the Second Law of Thermodynamics?
For an atmosphere opaque to long-wave radiation (f=1), my toy model of a multi-layer atmosphere predicts a surface temperature given by T₀⁴ = (S/σ)⋅(N+1). This has no upper limit, as the number of layers in the atmosphere increases.
It appears that the model is predicting that, with a sufficiently thick long-wave-absorbing atmosphere, a planet could achieve a surface temperature hotter than the Sun. That would be a violation of the Second Law of Thermodynamics. That can’t happen in reality. So, what is going on here?
The solution is very simple. If a planetary surface gets sufficiently hot, the surface will start to emit more and more of its thermal radiation as short-wave radiation. That short-wave radiation will pass through the atmosphere unhindered, just like the incoming solar radiation did. So, once a planet becomes hot enough to emit short-wave radiation, it can efficiently cool its surface.
As a result, a very thick long-wave absorbing atmosphere can never warm a planetary surface to be as warm as the Sun.
* * *
The other imagined violation of the Second Law that some people worry about relates to energy flowing from a cooler heat reservoir (the atmosphere) to a warmer heat reservoir (the surface of the planet).
But, the Second Law doesn’t say no energy can flow from cooler to warmer. It simply requires that the heat flow (i.e., the net energy flow), must be from warmer to cooler. As illustrated in the heat flow illustrations (Figure 4 and Figure 6), even with energy recirculation, heat always flows from warmer to cooler.
There is no violation of the Second Law.
Saturation
One of the naïve arguments against the possibility of increasing CO₂ having an effect on climate involves arguing that “CO₂ fully absorbs radiation after it travels a relatively short distance through the atmosphere, so how could adding more CO₂ make any difference?”
My toy model offers some insights regarding this issue.
In the model, each layer is assumed to absorb 100% of the longwave radiation within the fraction f of wavelengths that are absorbed. Yet, despite this, each added layer of atmosphere increases the surface temperature.
Whether or not the atmosphere absorbs long-wave radiation many times over is irrelevant to the potential of increasing greenhouse gases to lead to more warming.
However, there is a different type of “saturation” that does have an element of reality.
The energy recycling fraction in my multi-layer model of the atmosphere is given by β = N⋅f/(N+1). For large N, this become approximately β ≈ f⋅(1 − 1/N). So, as the total number of long-wave-opaque layers (or equivalently, the concentration of greenhouse gases) increases, additional layers do have smaller and smaller impacts on the energy recycling fraction.
This is vaguely like the assertion that the impact of increasing CO₂ levels is logarithmic, so that you need to keep doubling CO₂ levels to get comparable changes.
But, the mathematical form for this “saturation” effect isn’t quite the same. Where does our toy model go wrong?
The toy multi-layer model assumes that various wavelengths of long-wave radiation are either 100% transmitted or 100% absorbed. Yet, for real gases, there is a continuum in to the degree to which various wavelengths are absorbed.
One way of thinking about it is that the number of “opaque layers” in the atmosphere varies with wavelength. So, even if increasing gas concentrations has little impact on one wavelength, it might have a significant impact at another wavelength.
Another way of thinking about it is that, as you increase the concentration of long-wave absorbing gases, you effectively increase f, the fraction of wavelengths for which outbound long-wave radiation will be absorbed.
So, it makes sense that as you increase the concentration of long-wave-absorbing gases, the impact of additional increases declines, but in principle, there will still be an impact.
How many “layers” does Earth’s atmosphere have?
I calculated how an atmosphere with only radiative heat-transfer might affect surface temperature, as a function of how many “layers” the atmosphere has.
But, how do we decide how many layers there are in an atmosphere, for purposes of applying this model?
Recall that, in discussing Figure 1, I said that the formulas involved require one to assume that (a) the layer has sufficient optical depth that it absorbs most of the incident radiative at the wavelengths of interest and (b) the temperature doesn’t vary much across the layer.
This is a rough model, and there is no hard number one what constitutes absorbing “most” of incident radiation. But, an optical depth 2 would absorb 86% of incident radiation, so maybe that would be the minimum optical depth we’d want to associate with a layer?
For radiation with a wavelength of 15 microns, where CO₂ absorption peaks, the optical depth of Earth’s atmosphere may be around 100. So, that might suggest the use of as many as 50 layers in our toy model. But, at a wavelength of 14 or 16 microns, the optical depth is around 10, corresponding to no more than 5 layers. (However, if you divided the atmosphere into only 5 layers, the assumption that temperature doesn’t vary much across a layer would be unlikely to be valid.)
In general, optical depth varies strongly with wavelength. And, for many wavelengths, atmospheric temperature can be expected to vary over the distances needed for full absorption of those wavelengths.
The bottom line is that the toy model cannot be expected to model the behavior of the real atmosphere. (Let’s not forget that the toy model omits convection, which makes it even more likely that it could quantitatively describe the real atmosphere.)
Real climate models make use some of the ideas I’ve included in the toy model, but they fill in an enormous number of details that I’ve left out.
Conclusions
How things play out in a real atmosphere is, of course, vastly more complicated than can described quantitatively by models as simple as a what I’ve offered here.
Yet, a simple model like what I’ve shared here can help illustrate general mechanisms, and clarify some otherwise mystifying phenomenon. These simple models explain things like how back-radiation fluxes can be larger than the absorbed solar flux, and how more atmospheric radiation can reach the surface than reaches space.
I hope this has been helpful.
Most of the energy “absorbed” by various so-called greenhouse gasses (GHGs) is NOT re-radiated (scattered) in all directions. It is transmitted by conduction (direct collision) almost immediately to increase the latent heat of neighboring air molecules, only a tiny portion of which are capable of re-radiating. Furthermore, all the radiating done by CO2 molecules in the air cannot be apportioned to simply equating to 100% absorption of ground based IR, but must be apportioned from two sources, the amount directly re-radiated PLUS the amount generated from conduction (direct collisions) with faster moving molecules of air that excite the symmetrical vibration of the so-called GHGs that were not previously excited. Furthermore, when convection has caused the warmed air to rise, and raised the elevation at which this re-radiation occurs, the down-welling portion of air that CAN radiate downward, now has a much thicker atmosphere to penetrate for that re-radiated energy to strike the ground. And we haven’t even begun to discuss the additionally confounding energy transfers de to evaporation and condensation which are also carried by gasses rising due to convection. Layers…. HAH! I spit on your layers.
Much of what you describe names things that happen, but you’re making it seem unnecessarily complicated. (Some of your terminology is off too. Gas molecular collisions aren’t generally referred to as “conduction”; that’s just part of how energy is thermalized in a gas. And “latent heat” is specifically about water vapor.)
The bottom line covering most of what you say is: radiation absorbed by CO₂ warms the mixed gas that contains the CO₂. A mixed gas containing CO₂ that is slightly warmer will radiate slightly more.
As for convection, yes, that does change where the atmosphere will have what temperature, which affects the dynamics of longwave radiation in some ways. A detailed climate model needs to account for that.
I don’t mean to be tiresome, and I understand that my use of the term ‘conduction’ in this argument might be considered unconventional. Nevertheless, thermalization of a mass, even a mass of air by direct collision IS exactly the definition of conduction. I’m sticking with it. I don’t know, but I think maybe you’re hinting somehow that conduction only occurs at the boundary layers between solid or liquid surface of the Earth, and the gas layer of the atmosphere. If so, no one should think that.
My main point is that the energy that moved away from the solid or liquid surface as 15um IR is REMOVED from the radiation ‘budget’ altogether, and is moved into the thermal ‘budget’ where most of the mass is O2 and N2 and convection rules the day, and massive amounts of energy, now in the form of latent heat, lifts itself away from the surface of the Earth by rising up and carrying the added potential energy of evaporated water with it. Quantum mechanically, I suppose it’s a very difficult exercise to predict what the surface temperature will be because of a few extra parts per million of CO2. But when we model for simplicity, we can clearly presume the null hypothesis to be that any difference will be damn close to zero. Because the extra CO2 just means that the existing energy of the surface will be more efficiently thermalized in a slightly narrower band above the surface, at which point, the warmer air rises up and away from the surface, just as it did before those extra molecules were added. No new energy is added into mix, because that’s all from the sun heating the surface during the daylight. The energy that warmed via IR, absorption and conduction was not trapped in CO2 molecules trading 15um IR energy with each other for all time.
Back radiation budget exercises IMO, turn CO2 scattered light into a sort of perpetual motion machine that both HEAT the N2 and O2 mass of the atmosphere, and yet still miraculously bounce back and re-warm the surface as IR radiation. NO! You can’t use it to do both things, you silly alarmists (this comment isn’t directed at Bob W.).
I don’t have any deep objection to your using the term “conduction” in the way that you do.
There is not a separate “radiation budget” and “thermal budget.” They are (almost) always inextricably linked. There is thermal energy, and sometimes that resides in matter, and sometimes it is transported by radiation.
A mixed gas at any finite temperature can transport heat via convection (of sensible or latent heat) and can also absorb or radiate longwave radiation.
If the concentration of greenhouse gases is low, the mixed gas will be weakly couple to longwave radiation, and if the concentration is high, the mixed gas will be more strongly coupled to longwave radiation.
That macro way of looking at the situation seems to capture all one needs to know, without worrying too much about what happens at the molecular level.
And, for simplicity, we could assume the world is at absolute zero. But, that wouldn’t get us far towards understanding reality.
Your “null hypothesis” is a speculation seemingly based only on intuition. One needs to run the numbers to know how big an effect there is likely to be.
That’s not all that it means.
How CO₂ affects the air near the surface is one of the least interesting things about its impact. Its impact is systemic and emergent.
One way of describing its impact is this. Increasing the concentration of CO₂ raises the altitude at which the atmosphere becomes transparent to radiation in the 14-16 micron band.
The temperature of gas at this altitude will (with some complications) be in radiative equilibrium with space, at some temperature Tx.
If you increase the concentration of CO₂, you increase the altitude where the atmosphere is at temperature Tx. Given a tendency of the atmosphere to have a generally increasing temperature with decreasing altitude, with a certain lapse rate, raising the altitude where temperature Tx occurs can lead to temperatures at all lower altitudes increasing, to maintain the same temperature profile.
This is just one example of a way in which the impact of increasing CO₂ concentrations need not have anything to do with its effect on near-surface air.
I get it. You really want to be sure that energy is being conserved. What you think you’re hearing sets off alarms, sounding like someone isn’t paying attention to energy conservation. You want no part of that.
Makes sense.
I wonder if you can look at the situation from other people’s perspective.
Some people pay extremely close attention to details. They’ve been very careful with the rigor of their analysis. Energy is clearly being conserved, and the Second Law of Thermodynamics is clearly be honored.
Yet other people keep showing up, saying “That doesn’t look right! You’re cheating! That’s obviously wrong!”
And many are never, ever, are willing to actually look at the details. Nor do they offer an alternative analysis of their own that rigorously and correctly attends to all the relevant details.
They simply refuse to believe that things can work in the way they interpret things being described. Yet, often, they’re consistently misunderstanding something that was said.
This creates a tragic impasse. Yuck.
If you’re willing to talk through the details, I’m happy to do that, to unpack the particular moment in the analysis where you think something incorrect is being said. Then we could unpack that. Maybe one, or both of us, might learn something.
* * *
In my energy-recycling diagrams, Figures 2, 3, and 4, energy conservation and the Second Law of Thermodynamics are both honored.
Energy fluxes from the surface to the atmosphere can be carried by convection or radiation. It doesn’t matter. Either process contributes to warming of the mixed-gas atmosphere. Some fraction of that warming leads to longwave radiation being emitted, which cools the air.
So, energy isn’t doing double-duty there. It’s either thermal energy in the air, or it’s radiation, but it’s never both.
Nice work, Bob. Here are a couple of comments:
1) My calculus students always hated infinite series. They wondered if anything real actually involved an infinite series. Here is a good example of something that does.
2) Any time a person tries to explain radiation transfer, the explanation becomes long winded and people stop paying attention. It is an irony, therefore, that to fully understand the Greenhouse effect one has to understand radiant transfer — we resort instead to very simple explanations.
3) In engineering terminology the Earth is a re-radiating panel with a special coating on it to make it warmer. It wasn’t designed this way, but it’s how things turned out. In this sense the Earth is like a solar panel intended to make warm water.
Infinite series is the reason I washed out of a ChE PhD program, and left with my MS!
I got most things the first time. I remember in calculus when infinite series did not compute, and that thought “when will I see infinite series again?”.
Answer was in the 4th semester of Grad School “Advanced ChE Mathematics”. It was a whole semester of solutions involving infinite series! I scored 16/100 on one off the tests!
Your simple model (and apparently W&M’s) ignores reflection of both the incident SW radiation and the recycled LW radiation.
Neither model ignores reflection of the incident short-wave radiation. It’s just not included as an explicit term, because it doesn’t need to be. What I call S is the absorbed solar flux, i.e., the portion of sunlight that is not reflected back to space.
As for longwave… the emissivity of the Earth’s surface is 0.94. That indicates that the surface reflects, on average, 6 percent of recycled longwave radiation. It’s true that my simple model does not take that into account.
That’s one more aspect of it being a “toy model”, which includes only certain aspects of the real physics. It’s one of many simplifications.
ScarletMacaw
Jesus what a trivial reaction…
J.-P. D.
Bob, you said:
Strictly speaking, the long-wave fraction that is potentially re-absorbed by the surface is only 1/2 of the total for a flat Earth with an infinite extent. Because the surface is curved, some of the ‘downward’ radiation will be tangent to the surface of the Earth and escape to space (minus any atmospheric absorption). The higher the radiating layer, the larger the fraction of the downward radiation that will escape to space.
This isn’t critical to your argument, but it does make things a little more realistic if one acknowledges that the downward fraction is an upper-bound. It also demonstrates that the correct calculations require an integration that takes into consideration the altitude of the various hypothetical layers. That is, the assumption of the sum of the infinite series is too high and the equilibrium point is different than assumed.
Clyde Spencer
Maybe you think, before writing, a bit about the fact that the highest altitude for CO2’s activity isn’t 2000 km above surface, but… merely 50 km.
And that therefore, the incidence angle of backradiation reaching Earth is near 180 °, and hence the amount of backradiation around Earth, thus reaching outer space, is absolutely negligible.
Thanks for doing the simple trig math at home 🙂
J.-P. D.
Yes, the cone of depression is only about 1/2 a degree. We can quibble about the meaning and justification of calling a small angle “absolutely negligible.” It is calculable and is about one part in 360. Would you be upset if you had to spend one day a year in prison? Would you complain if your paycheck was 1/360th smaller than what the company had promised to pay you?
I offered a refinement to the statement of the problem where I acknowledged “This isn’t critical to your argument, …”
Maybe next time you should do “the simple trig math at home” before commenting.
Bob Wentworth
Thanks for your excellent contribution, especially for your patient, qualified replies to many opponents.
I’ll have to spend lots of more time into it.
You head post reminds me an article written in 2011 by two French scientists, Jean-Louis Dufresne and Jacques Treiner:
https://www.researchgate.net/publication/275205925_L'effet_de_serre_atmospherique_plus_subtil_qu'on_ne_le_croit
Unfortunately, it has been written in French, and Adobe is doing its best to prevent us from easily transferring parts of PDF documents into Google Translate’s windows, for example.
A simplified resumee exists in pasteable HTML form however:
https://www.centrale-energie.fr/spip/spip.php?article151
Merci beaucoup
J.-P. D.
Merci.
I appreciate the reference. I haven’t taken the time to try to translate, but it does appear that it covers some similar territory, and more.
🙂
Bob
Job
Take an imaginary Earth model with no CO2 produced r H2O in its atmosphere say only nitrogen.Compared to real, this would still be heated by solar radiation. Without radiative gases to cool it, what temperature would it reach?
Seems to me that you have to include nitrogen and oxygen as able to conduct and lose heat. Geoff S
Without any greenhouse gases, the atmosphere would have very little capacity to cool the surface.
I recently realized, that without greenhouse gases there wouldn’t be any convection. (Greenhouse gases provide the elevated “heat sink” needed to provide a “thermal driving head” to power natural convection.)
So, the surface could transmit heat to the atmosphere only via conduction.
But, without greenhouse gases, the atmosphere would have no means of cooling itself. (It couldn’t radiate any power to space.)
So, an atmosphere without greenhouse gases would have only minor effects on the temperature of the surface, compared to the temperature it would experience in a vacuum.
Probably the most important effect it would have would derive from the way that atmospheric pressure increases the thermal conductivity of dirt.
The devil is in the details, not the higher order vague toy models.
A) CO₂ is not homogenous throughout the atmosphere.
The higher the elevation, the lower the content of CO₂. CO₂ has been known to pool and suffocate humans and wildlife. Without winds and storms to keep mixing the lower atmosphere, oxygen based life would be difficult.
B) CO₂ longwave radiative interactivity is miniscule.
C) CO₂ emissions suggest the molecules have reached a temperature where emissivity is possible.
None of this affects your toy models or atmospheric layers. It does affect any claims that atmospheric CO₂ prevents or reduces longwave radiation to space.
As long as one realizes that H₂O is the main driver for all of the longwave radiation absorption and emissions as depicted.
There are some serious problems with this model. For one, the “f” factor in Figure 5 is NOT a constant. Assuming that IR photons at a wavelength that can be absorbed by CO2 or water vapor are emitted at an intensity Io from the earth’s surface, the fraction of these photons absorbed is proportional to an absorbance coefficient times the number of molecules of GHG per cubic meter.
Even if the mole fraction Yc of CO2 in the atmosphere is assumed constant, the number of CO2 molecules per cubic meter is proportional to YcP/RT, where P is atmospheric pressure and T is absolute temperature. It is well known that atmospheric pressure decreases with altitude z, and at the adiabatic lapse rate, T(z) = To * [P(z)/Po]^[(k-1)/k], where k is the ratio of specific heats Cp/Cv of air, and To and Po are the temperature and pressure of air at the surface.
Since k for air is about 1.4, the absolute temperature at altitude proportional to absolute pressure to the 0.29 power, and the ratio YcP/RT decreases proportional to absolute pressure to the 0.71 power. This means that, for a GHG at a constant mole fraction throughout the atmosphere, more IR radiation is absorbed close to the surface of the earth than at higher altitudes, and “f” decreases with altitude.
Unlike CO2, water vapor (a far stronger IR absorber than CO2) is not evenly distributed in the atmosphere–it is at higher concentrations near the surface, but at low concentrations above any cloud layer. Absorption by water vapor in the lower part of the atmosphere can “screen out” IR radiation (at wavelengths absorbed by water vapor) so that it never encounters a CO2 molecule, which minimizes the relative effect of CO2. Over a tropical ocean, water vapor concentrations can be greater than 10,000 ppm (1 mol%), or more than 25 times CO2 concentrations at the surface.
Another problem is to determine how absorption of IR radiation by GHG molecules can warm the atmosphere. If a GHG molecule (CO2 or H2O) absorbs a photon of IR radiation, an electron is raised to a higher energy state, and one of two things can happen. If the molecule re-emits another photon of the same energy in a random direction, on average almost half the photons will radiate toward the earth, and the rest will radiate toward space, but the net kinetic and rotational energy of the GHG molecule is not changed, and on the macro scale, there is no change in temperature.
However, if a GHG molecule absorbs an IR photon, then collides with another molecule (such as N2 or O2, the most common molecules in air), it can transfer its additional kinetic energy to the other molecule, and NOT re-emit a photon. On a macro scale, this would increase the temperature of the layer in which the GHG molecule is located, and decrease the total flux of IR radiation to space, without any “back radiation” toward earth.
This effect would also result in the outward-bound heat flux Q decreasing with altitude.
Also, since the nitrogen and oxygen molecules in the atmosphere do not absorb nor emit IR radiation, the factor “f” has to be proportional to absorption coefficient times YcP/RT (plus a similar factor for water vapor), integrated over the part of the spectrum where CO2 and water vapor absorb IR radiation. Depending on where this range of wavelengths (or wave numbers) occurs on the Planck function, this may not be proportional to T^4.
In summary, the effect of additional CO2 on warming the atmosphere is based on the part of IR radiation that is not absorbed by water vapor, that is absorbed by CO2 molecules and not re-emitted, and that is not absorbed by CO2 molecules lower in the atmosphere.
It’s true that optical depth varies with altitude, so that for layers to have roughly similar effect, you’d need to use thicker layers at higher altitudes. The factor f is only partly about optical depth, it’s also about what wavelengths do and don’t absorb. Still, it’s true that f will vary somewhat.
Most of the details you name are addressed by the caveat “It’s a toy model, not intended to reproduce all the intricacies of the atmosphere.”
No. The relevant 15 micron CO₂ absorption band involves a transition to a molecular flexing vibrational state. No electron excitation is involved. Typically, the vibrating CO₂ molecule collides with another molecule in the mixed gas before it re-radiates. But collisions in the mixed gas are constantly leading to some CO₂ molecules vibrating, and being capable of radiating.
When a vibrating CO₂ molecule radiates, it stops vibrating, and this incrementally cools the mixed gas.
You’re looking at the details too finely, in a way that introduces errors in the description and the conclusions.
The bottom line is that if a mixed gas containing greenhouse gases absorbs a photon of radiation, it incrementally warms the gas. A warm mixed gas is always spontaneously radiating in all directions, and it radiates slightly more when it is slightly warmer. Radiating a photon incrementally cools the mixed gas.
Well, what you describe only has this effect because systemic errors in the description create this outcome.
In equilibrium, it cannot be that the average, net outward-bound heat flux Q changes with altitude (summing over all modes of heat transport), except to the extent that some insolation is absorbed in the atmosphere. (This atmospheric absorption means that the average, net outward-bound heat flux Q must increase somewhat with altitude.)
Well, in the toy model, the layers are assumed to be sufficiently optically thick that the incremental absorption per unit length no longer matters.
But, ultimately, “f” and “N” are a little too simplistic to capture all the details of the radiative physics.
Agreed. Atmospheric radiation scaling as T⁴ is a crude over-simplification.
Not actually true.
Consider the results of my N-layer toy model. In that model, every layer is assumed to be entirely opaque to the wavelengths that are absorbed. So, for layers 2 through N, all the longwave radiation from the surface that could be absorbed has already been absorbed. Yet, layers 2 through N still contribute to warming of the surface.
Today’s scientists have substituted mathematics for experiment, and they wander off through equation after equation, and eventually build a structure which has no relation to reality. N.Tesla couldn’t be more right.
[[A particular layer of the atmosphere is assumed to be at a temperature, T₁. This temperature is the temperature that equalizes the flows of energy entering and leaving that layer. According to the diagram, the layer will emit long-wave radiant energy equally in all directions, with a flux fσT₁⁴ being sent upward and an equal flux being sent downward.]]
This paragraph along with the whole nonsense diatribe is pure dodo climate science.
You can immediately throw it in the trash when you see imaginary atmospheric layers with T^4 arrows coming out of them up and down. This is total ignorance, because gases don’t emit black body radiation, only photon by photon radiation at specific wavelengths based on absorption and reemission. A black body must be solid or liquid, and its number one power is the ability to absorb and emit radiation at all wavelengths. In the Earth-Sun climate system, only the Sun and the Earth are black bodies. The atmosphere doesn’t qualify. Their total power output over all wavelengths comes nowhere close to anything times T^4, and indeed doesn’t depend on T, only the incident photons.
End of this dodo line of reasoning.
I already blocked Wentworth on Quora after finding him pushing every IPCC hoax and then denying any connection with them while coming back over and over palming himself off as an expert. It didn’t take long to find out that his understanding of radiative physics is nil, especially when he failed to even mention key laws until I taught him.
Who am I? Go ahead and check me out. Do you believe that Wentworth has any credibility? Would you like to learn real climate science that destroys every IPCC lie? The only place to go is my free Climate Science 101 course that I spent years perfecting to cover every key physical law methodically and answer all objections. Until you master it you’ll always be a climate science dodo, open to IPCC-twisted brain manure dumps like this one.
http://www.historyscoper.com/climatescience101.html
For anyone interested, I’ve written a fairly popular deconstruction of one of TL Winslow’s main ideas.
Not a deconstruction, an exposition of rank ignorance, like the insane claim that gases emit T^4 radiation.
No matter how many times I disprove his nonsense, it bounces off.
[[Notice that the radiation emitted by CO₂ is still centered around a wavelength of 15 μm. However, for CO₂ at 0ºC the total radiance emitted is 4.4 times as large as it is for CO₂ at -80ºC. Surely, radiation from CO₂ at 0ºC will have more warming effect than radiation from CO₂ at -80ºC given this more than four-fold increase in power!]]
Duh, -80C is -80C. Power is irrelevant. And what is 0C CO2? That’s a gas, not a solid. Atmospheric CO2 only reemits 15 micron photons absorbed from the surface, and has no total radiance as such unless you’re considering the entire air column. Is he trying to treat gas as solid again, claiming that CO2 spontaneously emits radiation based on its temperature? ROTFL.
He then claims that 15 micron radiation isn’t at -80C by regraphing Planck’s Radiation Law in terms of power/Hz instead of power/micron. He never ‘gets’ that it’s not power vs. Hz. vs. power vs. micron. The power/Hz curve is basically the slope of the power/micron curve, so of course its peak is way different, but Nature still puts most of its dry ice photons around 15 microns, and atmospheric CO2 still can only absorb and emit 15 micron photons one at a time, not in a T^4 Planck power-wavelength curve that has warming power.
Why did WUWT allow this charlatan to publish a whole article full of crackpot physics that licks IPCC butt? I hope there’s a good answer.
My opinions as a thermo-mechanical engineer is that you can’t neglect the main heat exchange mechanisms just because they are more complicate to understand than simple radiation between gray surfaces. Extract too much information from this toy model without the main mechanism of heat transfer in the troposphere at your own peril of remaining ignorant.
It is true that toy models are useful to further understandings of physics, but not every toy model furthers the understanding of atmospheric thermodynamics of planet Earth.
If you were honest you wold at least try to put a number on the amount of heat that is transported by convection to the cloud tops (caused mainly by raising vapor) in relation to the radiative fluxes. Of course this factor is not homogeneous changing greatly by latitud.
This purely radiative model is only accurate near the poles where there is (almost) no incoming radiation. In the tropics, where most of the incoming heat is absorbed the dominant mechanism is convection that transfers heat upwards into upper layers of the atmosphere and pole-wards.
The tropics are the key regions of earth atmosphere and the toy model does’t cut it there.
Radiation and convection are both very important heat transport mechanisms in the atmosphere.
I emphasized radiation in this post because that is one of the most widely misunderstood aspects of the Earth’s thermodynamic system.
You can’t build a correct understanding of the Earth’s thermodynamics if you understanding of the radiation piece is broken.
Yes, you ultimately need to consider it all.
* * *
By the way, I just recently came to understand that, in the absence of greenhouse gases, there wouldn’t be convection either. (Radiation by greenhouse gases provides the elevated heat sink needed to generate the “thermal head” needed to power natural convection.)
For what it’s worth, for Figures 2, 3, and 4 of my essay, convective effects are included.
But, as I’ve said, I was trying to improve understanding about what radiative energy recycling effects can and cannot do, not model the actual atmosphere.
The thermodynamics of the stratosphere are just as important as the thermodynamics of the troposphere in Earth’s thermoregulation.
And in the stratosphere, radiative effects are dominant.
This supposed back radiation mechanism that increases its own energy and increases its own temperature is a mathematical psychobabble describing an over unity perpetual machine like energy amplifier that catches back its own emitted radiation and ads it up to itself.
It is no different than eating a thousand calories lunch, then vomiting it, then eating it again, and then claiming you are getting fatter because you doubled you calories intake.
Good exercise of ideological mathematics with no connection to physical reality. How is that no one of us, poor thermal engineers, are able to design devices that take advantage from this stunning, 1st-Law-breaking, radiant energy recycling principle ? If this is science, how deep low has it fallen
These principles are very much used in engineering:
It only looks like there is “1st-Law-breaking” if you’re unwilling or unable to pay attention to details.
Just because something isn’t familiar to your way of thinking doesn’t making not real.
mmm…isn’t familiar to my of way of thinking ?
Please show us any solar panel that uses, say, 10 layers of separated glasses in order to recycle the long wave radiation and increase, by a factor of 1.5 (or whatever) its watts output. That would seem straightforward, if such ideological math would reflect real physics.
Or : how is that you don’t feel your face warming when standing in front of the mirror ? Your face radiates approximately 500 W/m2 and your mirror reflects it back to you. The wall around the mirror only provides 400 W/m2 (assuming it has a temperature of 18°C).
You should well feel that difference, when moving from wall to mirror !
With a 2 m2 mirror you would have a nice “recycling” heater, warming you with (500-400)*2 = 200 W, a nice gift coming from…yourself. Move yourself closer to the mirror…and feel the heat !
You are misunderstanding what is being claimed.
A barrier that passes shortwave but impedes longwave radiation leads to a temperature increase. This is relevant when building a “solar panel” of the sort designed to heat water.
A barrier that passes shortwave but impedes longwave radiation does not increase the power that can be extracted from the system.
Thanks for the interesting question.
First, let’s check if a mirror reflects thermal radiation. Most mirrors are made with aluminum coating the back side of a sheet of glass. Aluminum does reflect longwave radiation. Common glass is transparent to longwave radiation. So, I accept the premise that a mirror should reflect thermal radiation.
So, what happens if you stand in front of a mirror? If you stand 1 meter from a mirror, the effect is essentially the same as another person standing 2 meters away from you. I don’t know about you, but I can’t feel the warmth of another person’s body from that distance.
However, I just tried putting my face about 1 cm from a mirror, and I did feel warmth.
Then I tried other surfaces. Putting my face next to a white wall, I also felt some warmth. However, putting my face next to a dark brown piece of furniture, I felt no warmth. I infer that the white paint had significant ability to reflect longwave radiation, but the furniture did not.
In the normal geometries for using a mirror, the inverse-square law applies. Not much of your face’s thermal radiation is reflected back to it by a flat mirror unless you are very close to the mirror.
The predictions concerning a barrier impeding longwave radiation leading to noticeable warming generally apply to situations where that barrier covers all or most of the paths for longwave radiation to escape. That’s never going to be the case for a flat mirror on one side of you, even if it is held quite close.
However, a “space blanket” is an example of a barrier impeding longwave radiation which is typically used in a way that blocks most paths for longwave radiation to escape. “Space blankets” are specifically used to warm people, and, to a degree, they are effective in achieving this purpose. (Their effectiveness is reduced to the degree that they still allow convective air flow. And more layers of space blanket can be more effective, if one gets the air flow down enough that a better radiant barrier matters.)
In a more obscure and personal example, my father once worked on a project that involved a small plutonium heat source driving a Sterling cycle heat engine. The plutonium heat source was wrapped in many thin layers of radiant barrier. This provided very effective insulation, and led to a large temperature difference to drive the heat engine.
1) By increasing temperature, you are also increasing internal energy (U=m*Cp*DT), aren’t you ? Thermodynamics 101. If you increase energy, there is more energy that you can extract from the system.
2) Be honest with yourself. No, you don’t feel any warmer when close to the mirror. Go very close and you should feel it. You don’t. The mirror doesn’t heat you by radiating back your own heat. Try this experiment with two light-bulbs : two identical bulbs, except for the glass. One crystal clear, the other frosted. Frosted glass “traps” some of the light, heat, energy, inside. Crystal clear glass doesn’t. What is the effect of trapping this light ? Based on your theory, you would expect the filament to become hotter, so you should see it by the color temperature changing. It’doesnt change. The filament does not increase its temperature.
3) The problem with your belief is that you treat radiation like if it was convection or conduction. It isn’t. And blankets do not make you warmer by radiation (https://phzoe.com/2020/04/08/do-blankets-warm-you/).
4) I have not enough data to comment on the Plutonium heat source, but if what you claim is true, it seems likely that you are seeing the effects of convection/conduction, rather than radiation.
There is more energy that you could extract, transiently, but it’s not sustainable. The amount of energy that you could sustainably extract doesn’t automatically increase just because you increase the internal energy.
Suppose, for any type of conserved quantity (energy, mass, etc.) you have a flow rate 𝚽 into a system and an equal flow rate out of the system. The quantity of the conserved quantity that accumulates inside the system can be characterized as U = 𝚽⋅𝜏 where U is the quantity of that which is conserved, and 𝜏 is the mean time the stuff resides in the system. You can increase the amount of stuff by increasing that residence time. But, you can still sustainably only get stuff out of the system with a flow rate 𝚽.
As an example, consider a river that flows into a lake and then out again. How much water can you sustainable withdraw from the lake? You could take water out at a rate exceeding the flow rate for a little while, but that wouldn’t be a steady-state condition, and wouldn’t be sustainable. If you want a sustainable flow of energy, you cannot take water out of the lake at a faster rate than whatever is flowing in.
The rate at which you can remove water is limited by the flow rate in. But, that flow rate does not limit how much water can be accumulated in a lake (aside from issues of evaporation).
Do you believe that this is true for water?
It is mathematically identical to consider a flow of energy as it is to consider a flow of water. Energy flux into a system determines the rate one could sustainably withdraw, yet internal energy/temperature is not inherently determined by that energy flux alone. If you do something to increase the residence time of the energy in the system, you can increase internal energy/temperature. (That’s what’s going on with the “energy recycling” described in this blog post.)
I double-checked. Still happens the same way I reported.
The wattage of a filament is determined by its resistance. You can increase resistance either by making the filament longer or thinner. These have different implications regarding how much energy per unit area the filament radiates. This allows a manufacturer to tune the color temperature, independent of wattage.
I don’t accept the premise that one can obtain light bulbs that are truly identical except that one uses frosted glass and one doesn’t. The designs may look similar. But, the manufacturer can easily tune the filament design of the two bulbs to make the color temperatures equal, despite different thermal dynamics in the two cases.
In many ways, it is similar. They all convey heat across a temperature difference in a way that increases heat flux as the temperature difference increases:
Conduction: 𝚽 = ∆T⋅R where R is thermal resistance.
Convection: 𝚽 = h⋅(∆T)ᵇ where h is a heat transfer coefficient and b is a scaling parameter.
Radiation: 𝚽 = 𝜀𝜎F⋅((T₀ + ∆T)⁴ – T₀⁴) ≈ 4𝜀𝜎F⋅∆T [which needs to be tweaked if the materials are dissimilar]
The mathematics of heat flow is not dramatically different for these different modes of heat transfer.
What convinces you that they must be treated as totally different?
Go back to U = 𝚽⋅𝜏, as mentioned above. For a given energy flow rate, you can make something warmer by increasing the residence time of the energy. This is a general truth, and it doesn’t matter which heat flow mechanism you’re considering. (The link you reference is applying faulty logic.)
“Space blankets” are used as emergency blankets because they are effective at keeping people warmer. If it was just about convection, they could just use a film of plastic without bothering to make it reflective. They don’t. They make them reflective, because it increases their warming power.
Again, if it was about “the effects of convection/conduction” then there would be no reason for the designers to bother using reflective materials in the layers. Yet, they did use reflective materials.
Your posts keep becoming longer and longer, but I think you are missing the main point : empirical tests show your theory is wrong.
Mathematics is not bound by the laws of physics.
The two GE lamps I’m referring to are identical – according to the data sheet. The filaments are identical (type CC-8). The wattage are identical (200W). The color temperature are identical (2900K). The only difference is the light provided by the soft white glass that is lower (3405 lumen vs 3780 of the crystal clear) due to glass absortion. Yet filament temperature does not change by 1 degree.
“Backradiation” alone – if at all exists – isn’t able to heat-up the emitting source. I leave that to theorethical physicists to explain why. It should not be that difficult. Radiation is inherently different than other heat exhange mechanisms.
You can do the experiment using GE lamps models 16069 and 15585. I can send you the product datasheets if you need them.
It sounds like things seem pretty clear to you, and it would be nice if I wasn’t seemingly so dense and unable to see things the way that you do.
That sounds frustrating.
I’m doing my best to understand.
It’s frustrating to me as well, that you haven’t been able to see my perspective.
I’d like to figure out why we are seeing things differently, so that maybe we can achieve some sort of shared reality.
Towards that end, I appreciate you offering a concrete example.
It was fun to work through the math for the relevant physics. The prediction is that the filament in the frosted bulb should be 0.49K warmer than the filament in the clear bulb.
Since the manufacturer reports color temperature only to the nearest degree (and I suspect the actual precision is coarser than that), this prediction is consistent with the available data.
The light bulb example does not support the assertion that “empirical tests show your theory is wrong.”
* * *
If you’re certain that radiative heat-trapping cannot lead to warming, I’m guessing that’s pretty frustrating to hear.
Maybe you could find a different example that would support your hypothesis?
I’ve sometimes been convinced I was wrong. I publicly announced that I was wrong about an analysis of mine in a recent blog post.
I’m willing to be convinced, based on the evidence. Are you?
That’s true for mathematics in general. But, the laws of physics have been codified in mathematical form. The mathematical formulation of the laws of thermodynamics has been tested empirically for over 150 years and have never, even once, been found to be wrong.
This is equally true for radiative heat transfer as for other portions of thermodynamics.
(Sometimes erroneous calculations have been done, but that’s different than the math or physics themselves being wrong.)
As a theoretical physicist, I regret to tell you that you seem to be mistaken in your belief (that “‘Backradiation’ alone… isn’t able to heat-up the emitting source”), based on everything I understand about physics.
As for doubts about whether backradiation “at all exists,” it’s not rocket science to measure it. Here’s one paper that shows (see Figure 3) the data for atmospheric downwelling longwave radiation as a function of time of day and month of year, based on measurements done in Oklahoma.
The light bulb example led me to some physics that I hadn’t previously thought about. That relates to the importance of the “view factor” in radiative heat transfer.
In the case of a light bulb, the radiation emitted by the filament is 100% “viewed” by the glass bulb. But, for the radiation emitted by the interior of the glass bulb, only 0.37% reaches the filament (based on an analysis of the manufacturer’s specifications for the particular bulbs in question).
It is this difference in “view factor” that leads to the frosted glass bulb being not very efficient in warming the filament.
A light bulb and the Earth are different in this regard. The “view factor” between the atmosphere and the surface of the Earth is essentially the same (100%) in both directions. This geometric difference means that atmospheric heat trapping is relatively more efficient at warming the surface, while heat trapping by a frosted light bulb is relatively less efficient at warming the filament.
Good point, I agree with your conclusion that due to areas/view factors ratio, the “back radiation” filament heating (if existed) would be almost meaningless anyways. However here are a couple of experiments that arrive to the same result in a much more cogent way :
The error of your theory is that your found “mathematically compliant” solution is “unstable”. From 1st LOT dU=mCpDt=Q+W, steady state (equilibrium) means dU=0 => dT=0. Since obviously W=0 (no work being done) then this involves Q=0 i.e. the two “plates” tend to an equilibrium condition with zero heat exchange (Q=0) between the two, just like when you put in contact two pieces at different temperature. The colder body, will tend to increase its temperature until it becomes equal to that of the hotter heated body.
I’ll take the parts of your comment one at a time, since there is some work involved in digesting each part.
Here’s a slightly better site with the same Greenplate experiment report. (The other site has some broken graphics, in my browser.)
I wish that people “debunking” a theory would take the time to figure out what the theory they are trying to disprove actually predicts.
The theory predicts that, in the 10-minute run of the experiment, Arrangement 6 should produce a higher temperature than Arrangement 5 by a little less than a degree. That appears to be entirely consistent with the measured results.
My spreadsheet simulation of the experiment shows that the temperature of the second plate strongly lags the temperature of first plate. As a result, it takes around an hour to get to the stage where the second plate is adding 25℃ to the temperature of the first plate.
The experimenter ran the experiment for too short a period. As a result, the falsification is false.
The “experiment on back-radiation” demonstrates painfully confused thinking.
There doesn’t seem to be anything particularly surprising in the experimental results. Yet the author has the odd idea that somehow he has disproved mainstream understandings of the impact of back-radiation.
He writes:
These words are a nonsequitur because the author falsely equates “back-radiation results in the surface being warmer than it would otherwise be” or “back-radiation is absorbed by the surface” with “thermal radiation from the atmosphere is larger than thermal radiation from the surface.”
The latter is not what anyone claims to be true. So, falsifying this proves nothing.
He also writes:
Again, this is a nonsequitur, asserting that back-radiation “warming up the surface” has an implication that it doesn’t have, then declaring victory when that falsely-attributed prediction is falsified.
* * *
All the author has proved is that he doesn’t understand what is meant by the idea of back-radiation contributing to the surface being warmer than it would otherwise be.
I assume you’re talking about the “green plate” experiment?
Let’s pretend it’s a 1-dimensional problem, pretending the heat lamp is a little more all-encompassing of the front of Plate 1 than it actually is.
Then, the two cases in the experiment have heat flow that looks like this:
CASE A:
[Heat source] == Q ==> [Plate 1] == Q ==> [Heat sink]
CASE B:
[Heat source] == Q ==> [Plate 1] == Q ==> [Plate 2] == Q ==> [Heat sink]
The heat source is the heat lamp. The heat sink is the environment surrounding the apparatus.
Yes, the total dQ for Plate 1 is zero, but this is a result of a +dQ from the heat source and -dQ leaving Plate 1.
In Case B, there is a net heat flow from Plate 1 to Plate 2.
In Case A, the heat flow out of Plate 1 is σ(T1⁴ – Te⁴).
In Case B, the heat flow out of Plate 1 is σ(T1⁴ – T2⁴).
Because T2 > Te, it is necessary for T1 to be larger in Case B than in Case A in order to achieve the same heat flow.
QED.
In the depicted scenario, that can’t happen, because then Plate 1 would not be able to get rid of its heat through Plate 2.
As I’m guessing you will have gathered, from my perspective, each of the “experiments that arrive to the same result” were deeply flawed, and didn’t justify the conclusions they arrived at.
Where does that leave us?
I come back to the observation that reflective membranes are used as “emergency blankets” and as an integral part of building insulation.
As well as the fact that radiative heat transfer has been understood for nearly 200 years and the principles of analyzing radiating systems are extremely well-established and well-tested.
But, I suppose “your mileage may vary.”
Thoughts?
It seems what is of most interest is what the thermal profile of the atmosphere is at the “top of the atmosphere”, by which I mean the level at which most of the long wave radiation emitted upward escapes without being re-absorbed. Since there are many different greenhouse gases that level will vary by the wavelengths of interest but CO2 is one of the highest significant absorbers for longwave radiation so that is worth focusing on.